We have to prove:
It is equivalent to proving: So we are done if we are able to prove this.
now,
if is a positive real number,
if then (obvious)
if then (obvious)
if then so, (multiplying both sides by a negative number i.e changes the sign)
so for all case,
similarly,
summing these we get:
so we will be done if we prove or
by the AM-GM inequality, # as
proved (Rock)